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A maximum-exposure constraint in paired Raman acquisition bounds each detector frame so that a bright channel stays below saturation. For $n$ pump-on/off pairs with common per-frame cap $t_{\max }$, the total live exposures satisfyFor a fixed pair count and wall-clock budget, readout overhead in Raman exposure allocation leaves $T=T_{\mathrm{on}}+T_{\mathrm{off}}=W-2nd$. Using the full budget is feasible only if $0<T\le 2n{t}_{\max }$; otherwise one must change the pair count, shorten the acquisition or allow unused time.

Under the independent large-count shot-noise model, minimize $V=\alpha /T_{\mathrm{on}}+\beta /T_{\mathrm{off}}$. The feasible on-exposure interval isThe objective is strictly convex for positive on/off exposures. Its unconstrained optimum is $T\sqrt{\alpha }/(\sqrt{\alpha }+\sqrt{\beta })$; clip that value to the feasible interval, retaining positive time in both states. This follows from the derivative changing sign once, rather than requiring a new unconstrained allocation rule.

For an original model, set $W=1\mathrm{s}$, $d=0.010\mathrm{s}$, $t_{\max }=0.097\mathrm{s}$, $\alpha =2\times 10^{-6}\mathrm{s}$ and $\beta =0.5\times 10^{-6}\mathrm{s}$. With six pairs, $T=0.88\mathrm{s}$ and the unconstrained on total $0.58667\mathrm{s}$ exceeds its cap $0.582\mathrm{s}$. The constrained choice is $T_{\mathrm{on}}=0.582\mathrm{s}$, $T_{\mathrm{off}}=0.298\mathrm{s}$, with $V\approx 5.1143\times 10^{-6}$. Five and seven pairs give approximately $5.3285\times 10^{-6}$ and $5.2326\times 10^{-6}$ respectively under the same model. Six is preferable among these three choices.

The integer frame-count choice under a Raman saturation limit must also include switching cadence, drift and read noise when relevant. The primary apparatus used 97 ms exposures to avoid detector saturation and 10 ms shuttered readout.^[ref] Those values motivate the stipulated numerical cap above; they are not universal detector specifications. A different optical flux or detector gain changes the usable exposure limit.

Wall-clock slack in saturation-limited Raman frame allocation is time that remains unused when a fixed number of detector frames has already reached its permitted exposure caps. With $n$ pump-on/off pairs, per-frame cap $c$ and readout time $d$ per frame,assuming $W>2nd$. The integer frame-count choice under a Raman saturation limit compares these possibilities over different pair counts. Slack at fixed $n$ does not imply that the continuous exposure allocation was optimized incorrectly.

If $S_n>0$, both states can use their full live totals $nc$, and the shot-noise relative variance is $V_n=(\alpha +\beta )/(nc)$. Increasing only the wall-clock allowance cannot improve it: the marginal value of unused time below a Raman frame cap is zero in this capped regime. Another frame pair changes both the overhead and the allowable division of live time.

For an original example, set $W=0.50\mathrm{s}$, $n=2$, $c=0.10\mathrm{s}$ and $d=0.020\mathrm{s}$. The live time is $0.40\mathrm{s}$, readout uses $0.08\mathrm{s}$, and slack is $0.02\mathrm{s}$. With $\alpha =2\times 10^{-6}\mathrm{s}$ and $\beta =0.5\times 10^{-6}\mathrm{s}$, equal capped totals $0.20\mathrm{s}$ give $V_2=12.5\times 10^{-6}$.

Three pairs increase readout time to $0.12\mathrm{s}$, leaving only $0.38\mathrm{s}$ live time. Yet the larger per-state cap permits the preferred ratio two to one: $T_{\mathrm{on}}\approx 0.2533\mathrm{s}$ and $T_{\mathrm{off}}\approx 0.1267\mathrm{s}$. Their variance is $V_3=4.5\times 10^{-6}/0.38\approx 11.8421\times 10^{-6}$, lower despite less total live time. Relieving the weak state's cap outweighs the extra readout cost in this chosen model.

The primary Raman apparatus limited exposures to avoid detector saturation and alternated pump state during shuttered readout.^[ref] The timings above are separate illustrative choices. Read noise, sample drift and switching constraints can change the comparison; unused time and useful precision are different optimization quantities.